Hello,
Just to be sure: AM=Armenia and FI=Finland, right?

 

Yes, you have dual nationality French and US. But for a clear understanding, do you mean that you would like to pursue a doctorate in FRANCE where we would like to settle down?

 

So regarding your initial question, if your plan is to settle down in the US after your studies, it all depends of the type of relationship with your partner to be considered before you move (civil partnership or marriage?).
Otherwise, it means that you have to consider if your partner will apply for a work or student visa to follow you as he will have to do the paperwork independently.